On the Whitney extension property for continuously differentiable horizontal curves in sub-Riemannian manifolds

On the Whitney extension property for continuously differentiable horizontal curves in... In this article we study the validity of the Whitney $$C^1$$ C 1 extension property for horizontal curves in sub-Riemannian manifolds that satisfy a first-order Taylor expansion compatibility condition. We first consider the equiregular case, where we show that the extension property holds true whenever a suitable non-singularity property holds for the endpoint map on the Carnot groups obtained by nilpotent approximation. We then discuss the case of sub-Riemannian manifolds with singular points and we show that all step-2 manifolds satisfy the $$C^1$$ C 1 extension property. We conclude by showing that the $$C^1$$ C 1 extension property implies a Lusin-like approximation theorem for horizontal curves on sub-Riemannian manifolds. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

On the Whitney extension property for continuously differentiable horizontal curves in sub-Riemannian manifolds

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2018 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-018-1336-8
Publisher site
See Article on Publisher Site

Abstract

In this article we study the validity of the Whitney $$C^1$$ C 1 extension property for horizontal curves in sub-Riemannian manifolds that satisfy a first-order Taylor expansion compatibility condition. We first consider the equiregular case, where we show that the extension property holds true whenever a suitable non-singularity property holds for the endpoint map on the Carnot groups obtained by nilpotent approximation. We then discuss the case of sub-Riemannian manifolds with singular points and we show that all step-2 manifolds satisfy the $$C^1$$ C 1 extension property. We conclude by showing that the $$C^1$$ C 1 extension property implies a Lusin-like approximation theorem for horizontal curves on sub-Riemannian manifolds.

Journal

Calculus of Variations and Partial Differential EquationsSpringer Journals

Published: Mar 12, 2018

References

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